pod_basis_flow/pod_basis_flow.html

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      POD_BASIS_FLOW - PDE Model Reduction
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      POD_BASIS_FLOW <br> PDE Model Reduction
    </h1>

    <hr>

    <p>
      <b>POD_BASIS_FLOW</b>
      is a FORTRAN90 program which
      applies the principal
      orthogonal direction (POD) analysis to a set of solutions of a
      PDE that models two dimensional time dependent fluid flow.
    </p>

    <p>
      This procedure, originally devised by Karl Pearson, has arisen
      repeatedly in a variety of fields, and hence is known under
      various names, including:
      <ul>
        <li>
          the Hotelling transform;
        </li>
        <li>
          the discrete Karhunen-Loeve transform (KLT)
        </li>
        <li>
          Principal Component Analysis (PCA)
        </li>
        <li>
          Principal Orthogonal Direction (POD)
        </li>
        <li>
          Proper Orthogonal Decomposition (POD)
        </li>
        <li>
          Singular Value Decomposition (SVD)
        </li>
      </ul>
    </p>

    <p>
      <i>
        You almost certainly should rather use the simpler program
        called
        <a href = "../../f_src/svd_basis/svd_basis.html">SVD_BASIS</a>,
        which does not assume that the data comes from a particular
        fluid flow problem!
      </i>
    </p>

    <p>
      The purpose of the algorithm is to extract the dominant modes
      of behavior of the system; these modes could then be used, for
      instance, in a reduced order model of the physical system.
    </p>

    <p>
      For the computations considered here, a partial differential
      equation (PDE) has been defined, specifying the time-dependent
      flow of a fluid through a region.  The PDE specification includes
      a parameter <b>alpha</b> whose value strongly affects the behavior of
      the flow.  The steady state solution X0 is computed for a particular
      value of <b>alpha</b>.  Then the time-dependent problem is solved over a
      fixed time interval, with <b>alpha</b> varying from time to time.
      A set of several hundred solutions X(T(I),<b>alpha</b>(I)) are saved.
    </p>

    <p>
      The need is to try to extract from this solution data the
      typical modes of behavior of the solution, that is, a small set
      of orthogonal vectors <b>V</b> such that "most" of the solution
      vectors <b>X</b> can be well represented by a linear combination
      of elements of <b>V</b>.  Such a set of modes may then be used as
      a finite element basis that is highly tuned to the physics of the
      problem, so that a very small set of basis functions can be used
      to closely approximate the behavior of the solution over a range
      of values of <b>alpha</b>.
    </p>

    <p>
      The method of extracting information from the solution data
      begins by constructing an <b>M</b> by <b>N</b> matrix <b>A</b>, each
      of whose columns is one of the solution vectors <b>X</b>.  Thus,
      <blockquote><b>
        A = [ X1 | X2 | ... | XN ]
      </b></blockquote>
    </p>

    <p>
      Then the singular value decomposition of <b>A</b>:
      <blockquote><b>
        A = U * S * V'
      </b></blockquote>
      is determined using the DGESVD routine from the linear algebra package
      <a href = "../lapack/lapack.html">LAPACK</a>.
      A subset of the columns of the orthogonal <b>M</b> by <b>M</b>
      matrix <b>U</b>, associated with the largest singular values <b>S</b>,
      is chosen to form the POD basis.
    </p>

    <p>
      Because the data comes from a finite element computation, and
      the results may be used as a new reduced basis, it may be
      desirable to carry out mass matrix preconditioning of the data,
      so that output POD vectors are orthogonal
      in the L2 inner product (integration of the product of the finite
      element functions over the domain).
    </p>

    <p>
      The current version of the program assumes that a steady state
      solution <b>SS</b> of the PDE is known, and that a multiple
      of SS is to be subtracted from each solution vector before processing.
    </p>

    <p>
      <b>FILES</b>: the program assumes the existence of the following files:
      (the actual names of the files are specified by the user at run time.
      The names used here are just suggestions.)
      <ul>
        <li>
          <i>xy.txt</i>, contains the (x,y) coordinates of each node, with
          one pair of coordinates per line of the file;
        </li>
        <li>
          <i>ss.txt</i>, contains the (u,v) values at each node for the
          steady state solution, with one pair of values per line of the file;
        </li>
        <li>
          <i>uv01.txt</i>, <i>uv02.txt</i>, ..., contains the (u,v) values
          at each node for solution 1, 2, and so on;
        </li>
        <li>
          <i>element.txt</i>, contains the indices of the six nodes that
          make up each element, with one set of six indices per line of
          the file <i>(only needed if mass matrix
          preconditioning is used)</i>;
        </li>
      </ul>
    </p>

    <p>
      <b>INPUT</b>: at run time, the user specifies:
      <ul>
        <li>
          <i>run_type</i> describes how we subtract off the steady state,
          whether we drop some data, and other options.  The current
          values range from 1 to 8.  The most common value is 6, used
          with the TCELL data:
          <ol>
            <li>
              no steady state file, no preprocessing;
            </li>
            <li>
              no steady state file, no preprocessing;
            </li>
            <li>
              subtract 1/3 SS from solution 1, 5/3 SS from solutions
              2 to 201, and 1/3 SS from solutions 202 through 401.
            </li>
            <li>
              subtract 1/3 SS from solution 1, 5/3 SS from solutions
              2 to 201, and 1/3 SS from solutions 202 through 401,
              and drop the even-numbered data.
            </li>
            <li>
              subtract 1/3 SS from solution 1, 5/3 SS from solutions
              2 to 201, and 1/3 SS from solutions 202 through 401,
              and skip half the data and normalize it.
            </li>
            <li>
              subtract 5/3 SS from solutions
              1 to 250, and 1/3 SS from solutions 251 through 500, do not
              normalize.
            </li>
            <li>
              subtract 5/3 SS from solutions
              1 to 250, and 1/3 SS from solutions 251 through 500,
              normalize the data.
            </li>
            <li>
              subtract 5/3 SS from solutions
              1 to 250, and 1/3 SS from solutions 251 through 500, then
              drop the odd-numbered data, do not
              normalize.
            </li>
          </ol>
        </li>
        <li>
          <i>basis_num</i>, the requested basis size.
        </li>
        <li>
          <i>xy_file</i>, the name of the xy file.
        </li>
        <li>
          <i>steady_file</i>, the name of the steady state solution file
          (only if <b>run_type</b> is 1 or 2).
        </li>
        <li>
          <i>uv0_file</i>, the name of the first solution file (the program
          will assume all the files are numbered consecutively).
        </li>
        <li>
          <i>element_file</i>, the name of the element file, if mass matrix
          preconditioning is desired, or else "none".
        </li>
      </ul>
    </p>

    <p>
      <b>OUTPUT</b>: the program computes <b>basis_num</b> basis vectors.
      The first vector is written to the file <i>pod_001.txt</i>; again,
      the output vectors are written with two values per line, since
      this represents the two components of velocity at a particular
      node.
    </p>

    <h3 align = "center">
      Licensing:
    </h3>

    <p>
      The computer code and data files described and made available on this web page
      are distributed under
      <a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
    </p>

    <h3 align = "center">
      Related Data and Programs:
    </h3>

    <p>
      <a href = "../../m_src/brain_sensor_pod/brain_sensor_pod.html">
      BRAIN_SENSOR_POD</a>,
      a MATLAB program which
      applies the method of Proper Orthogonal Decomposition
      to seek underlying patterns in sets of 40 sensor readings of
      brain activity.
    </p>

    <p>
      <a href = "../../f_src/cvt_basis_flow/cvt_basis_flow.html">
      CVT_BASIS_FLOW</a>,
      a FORTRAN90 program which
      is similar to POD_BASIS_FLOW but uses CVT techniques to
      do the analysis of the data.
    </p>

    <p>
      <a href = "../../f_src/lapack_examples/lapack_examples.html">
      LAPACK_EXAMPLES</a>,
      a FORTRAN90 program which
      demonstrates the use of the LAPACK linear algebra library.
    </p>

    <p>
      <a href = "../../f_src/svd_basis/svd_basis.html">
      SVD_BASIS</a>,
      a FORTRAN90 program which
      is a simpler version of this program; it does not assume that
      the underlying data represents solutions of a fluid flow problem.
    </p>

    <h3 align = "center">
      Reference:
    </h3>

    <p>
      <ol>
        <li>
          Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford,
          James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum,
          Sven Hammarling, Alan McKenney, Danny Sorensen,<br>
          LAPACK User's Guide,<br>
          Third Edition,<br>
          SIAM, 1999,<br>
          LC: QA76.73.F25L36
        </li>
        <li>
          John Burkardt, Max Gunzburger, Hyung-Chun Lee,<br>
          Centroidal Voronoi Tessellation-Based Reduced-Order
          Modelling of Complex Systems,<br>
          SIAM Journal on Scientific Computing,<br>
          Volume 28, Number 2, 2006, pages 459-484.
        </li>
        <li>
          Gal Berkooz, Philip Holmes, John Lumley,<br>
          The proper orthogonal decomposition in the analysis
          of turbulent flows,<br>
          Annual Review of Fluid Mechanics,<br>
          Volume 25, 1993, pages 539-575.
        </li>
        <li>
          Lawrence Sirovitch,<br>
          Turbulence and the dynamics of coherent structures, Parts I-III,<br>
          Quarterly of Applied Mathematics,<br>
          Volume XLV, Number 3, 1987, pages 561-590.
        </li>
      </ol>
    </p>

    <h3 align = "center">
      Source Code:
    </h3>

    <p>
      <ul>
        <li>
          <a href = "pod_basis_flow.f90">pod_basis_flow.f90</a>, the source code.
        </li>
        <li>
          <a href = "pod_basis_flow.sh">pod_basis_flow.sh</a>,
          commands to compile and load the source code.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      Examples and Tests:
    </h3>

    <p>
      PDE solution datasets you may copy include:
      <ul>
        <li>
          <a href = "../../datasets/cavity_flow/cavity_flow.html">
          CAVITY_FLOW</a>, the driven cavity;
        </li>
        <li>
          <a href = "../../datasets/inout_flow/inout_flow.html">
          INOUT_FLOW</a>, flow in and out of a chamber;
        </li>
        <li>
          <a href = "../../datasets/tcell_flow/tcell_flow.html">
          TCELL_FLOW</a>, flow through a T-cell;
        </li>
      </ul>
    </p>

    <p>
      Once the reduced basis set is computed, it is possible to
      set up new finite element problems in which the reduced basis
      set is used as the finite element basis.  To see an example
      of such a calculation, refer to
      <ul>
        <li>
          <a href = "../../f77_src/tcell/tcell.html">
          TCELL</a>, the program which generated the TCELL data;
        </li>
        <li>
          <a href = "../../f77_src/tcell_mass/tcell_mass.html">
          TCELL_MASS</a>, used to calculate the mass matrix;
        </li>
        <li>
          <a href = "../../m_src/tcell_rom/tcell_rom.html">
          TCELL_ROM</a>, the TCELL reduced order modeling program;
        </li>
      </ul>
    </p>

    <p>
      This program has been run with a number of different datasets,
      and with various requirements as to normalization and so on.
      The purpose of most of the runs is to find a generator set of
      given size.  The input and output of each run is stored in
      a separate subdirectory.
    </p>

    <p>
      Each run of the code is stored in a separate subdirectory.
      Available runs include:
      <ul>
        <li>
          <a href = "run_01/run_01.html">run_01</a>,
          getting 16 POD vectors from the CAVITY_FLOW data.
        </li>
        <li>
          <a href = "run_02/run_02.html">run_02</a>,
          getting 16 POD vectors from the INOUT_FLOW data.
        </li>
        <li>
          <a href = "run_03/run_03.html">run_03</a>,
          getting 16 POD vectors from the TCELL_FLOW data.
        </li>
        <li>
          <a href = "run_04/run_04.html">run_04</a>,
          getting 16 POD vectors from the CAVITY_FLOW data,
          with mass matrix preconditioning.
        </li>
        <li>
          <a href = "run_05/run_05.html">run_05</a>,
          getting 16 POD vectors from the INOUT_FLOW data,
          with mass matrix preconditioning.
        </li>
        <li>
          <a href = "run_06/run_06.html">run_06</a>,
          getting 16 POD vectors from the TCELL_FLOW data,
          with mass matrix preconditioning.
        </li>
        <li>
          <a href = "run_07/run_07.html">run_07</a>,
          getting 16 POD vectors from the CAVITY_FLOW data,
          with mass matrix preconditioning.  We drop the odd
          numbered data vectors.
        </li>
        <li>
          <a href = "run_08/run_08.html">run_08</a>,
          getting 16 POD vectors from the INOUT_FLOW data,
          with mass matrix preconditioning.  We drop the odd
          numbered data vectors.
        </li>
        <li>
          <a href = "run_09/run_09.html">run_09</a>,
          getting 16 POD vectors from the TCELL_FLOW data,
          with mass matrix preconditioning.  We drop the odd
          numbered data vectors.
        </li>
        <li>
          <a href = "run_10/run_10.html">run_10</a>,
          getting 16 POD vectors from the INOUT_FLOW2 data.
        </li>
        <li>
          <a href = "run_11/run_11.html">run_11</a>,
          getting 16 POD vectors from the INOUT_FLOW2 data,
          with mass matrix preconditioning.
        </li>
      </ul>
    </p>

    <h3 align = "center">
      List of Routines:
    </h3>

    <p>
      <ul>
        <li>
          <b>MAIN</b> is the main program for POD_BASIS_FLOW.
        </li>
        <li>
          <b>BANDWIDTH_DETERMINE</b> computes the lower bandwidth of a finite element matrix.
        </li>
        <li>
          <b>BASIS_WRITE</b> writes a basis vector to a file.
        </li>
        <li>
          <b>CH_CAP</b> capitalizes a single character.
        </li>
        <li>
          <b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
        </li>
        <li>
          <b>CH_IS_DIGIT</b> is TRUE if a character is a decimal digit.
        </li>
        <li>
          <b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
        </li>
        <li>
          <b>DATA_IVEC_READ</b> reads an dataset of integer vectors stored in a file.
        </li>
        <li>
          <b>DATA_D2_READ</b> reads pairs of double precision numbers stored in a file.
        </li>
        <li>
          <b>DATA_SIZE</b> counts the size of a data set stored in a file.
        </li>
        <li>
          <b>DBLT_CHECK</b> checks the dimensions of a banded lower triangular matrix.
        </li>
        <li>
          <b>DBLT_PRINT</b> prints a band lower triangular matrix.
        </li>
        <li>
          <b>DBLT_PRINT_SOME</b> prints some of a band lower triangular matrix.
        </li>
        <li>
          <b>DIGIT_INC</b> increments a decimal digit.
        </li>
        <li>
          <b>DIGIT_TO_CH</b> returns the character representation of a decimal digit.
        </li>
        <li>
          <b>DPBL_CHECK</b> checks the dimensions of a positive definite symmetric band matrix.
        </li>
        <li>
          <b>DPBL_PRINT</b> prints a symmetric banded matrix.
        </li>
        <li>
          <b>DPBL_PRINT_SOME</b> prints some of a symmetric banded matrix.
        </li>
        <li>
          <b>FILE_EXIST</b> reports whether a file exists.
        </li>
        <li>
          <b>FILE_NAME_INC</b> generates the next filename in a series.
        </li>
        <li>
          <b>GET_UNIT</b> returns a free FORTRAN unit number.
        </li>
        <li>
          <b>I4_INPUT</b> prints a prompt string and reads an integer from the user.
        </li>
        <li>
          <b>I4_RANGE_INPUT</b> reads a pair of integers from the user, representing a range.
        </li>
        <li>
          <b>MASS_MATRIX</b> computes the mass matrix.
        </li>
        <li>
          <b>NODE_T6</b> returns the basis nodes for a 6 node triangle.
        </li>
        <li>
          <b>R8VEC_PRINT</b> prints a double precision vector.
        </li>
        <li>
          <b>REFQBF</b> evaluates a reference element quadratic basis function.
        </li>
        <li>
          <b>S_EQI</b> is a case insensitive comparison of two strings for equality.
        </li>
        <li>
          <b>S_INPUT</b> prints a prompt string and reads a string from the user.
        </li>
        <li>
          <b>S_OF_I4</b> converts an integer to a left-justified string.
        </li>
        <li>
          <b>S_REP_CH</b> replaces all occurrences of one character by another.
        </li>
        <li>
          <b>S_TO_I4</b> reads an I4 from a string.
        </li>
        <li>
          <b>S_TO_R8</b> reads an R8 from a string.
        </li>
        <li>
          <b>S_WORD_COUNT</b> counts the number of "words" in a string.
        </li>
        <li>
          <b>SINGULAR_VECTORS</b> computes the desired singular values.
        </li>
        <li>
          <b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
        </li>
        <li>
          <b>TIMESTRING</b> writes the current YMDHMS date into a string.
        </li>
        <li>
          <b>TRIANGLE_UNIT_SET</b> sets weights and abscissas for quadrature in a unit triangle.
        </li>
      </ul>
    </p>

    <p>
      You can go up one level to <a href = "../f_src.html">
      the FORTRAN90 source codes</a>.
    </p>

    <hr>

    <i>
      Last revised on 17 July 2004.
    </i>

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